Nuprl Lemma : fill-type-up-0
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)]. ∀[u:{Gamma ⊢ _:(A)[0(𝕀)]}].
((app(fill-type-up(Gamma;A;cA); (u)p))[0(𝕀)] = u ∈ {Gamma ⊢ _:(A)[0(𝕀)]})
Proof
Definitions occuring in Statement :
fill-type-up: fill-type-up(Gamma;A;cA),
composition-op: Gamma ⊢ CompOp(A),
interval-0: 0(𝕀),
interval-type: 𝕀,
cubical-app: app(w; u),
csm-id-adjoin: [u],
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
csm-ap-term: (t)s,
fill-type-up: fill-type-up(Gamma;A;cA),
cubical-app: app(w; u),
cubical-lambda: (λb),
swap-interval: swap-interval(G;A),
csm-swap: csm-swap(G;A;B),
cubical-type: {X ⊢ _},
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-ap: (s)x,
cc-fst: p,
cc-adjoin-cube: (v;u),
cc-snd: q,
csm-ap-type: (AF)s,
csm+: tau+,
csm-adjoin: (s;u),
csm-id: 1(X),
csm-comp: G o F,
pi1: fst(t),
pi2: snd(t),
compose: f o g,
cube-context-adjoin: X.A,
cubical-term-at: u(a),
interval-type: 𝕀,
constant-cubical-type: (X),
squash: ↓T,
true: True,
prop: ℙ
Lemmas referenced :
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
cubical_set_cumulativity-i-j,
csm-id-adjoin_wf-interval-0,
cc-fst_wf,
cubical-term_wf,
cubical-type-cumulativity2,
composition-op_wf,
cubical-type_wf,
cubical_set_wf,
fill-type-up_wf,
cubical-app_wf_fun,
csm-ap-term_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
cube_set_restriction_pair_lemma,
interval-type-ap-morph,
dM-lift-0-sq,
cubical-term-at_wf,
filling_term_0,
face-0_wf,
csm+_wf_interval,
csm-face-0,
context-subset-term-0,
constrained-cubical-term-0,
csm+_wf,
csm-id-adjoin_wf,
csm-interval-type,
interval-0_wf,
cc-snd_wf,
csm-composition_wf,
cc-adjoin-cube_wf,
cube-set-restriction_wf,
nh-id_wf,
subtype_rel-equal,
cubical-type-at_wf,
cube-set-restriction-id,
csm-ap-term-at,
cubical_type_at_pair_lemma,
squash_wf,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
hypothesis,
applyEquality,
because_Cache,
sqequalRule,
universeIsType,
inhabitedIsType,
lambdaFormation_alt,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
functionExtensionality,
independent_isectElimination,
setElimination,
rename,
productElimination,
Error :memTop,
applyLambdaEquality,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cA:Gamma.\mBbbI{} \mvdash{} CompOp(A)]. \mforall{}[u:\{Gamma \mvdash{} \_:(A)[0(\mBbbI{})]\}].
((app(fill-type-up(Gamma;A;cA); (u)p))[0(\mBbbI{})] = u)
Date html generated:
2020_05_20-PM-04_54_50
Last ObjectModification:
2020_04_13-PM-02_56_43
Theory : cubical!type!theory
Home
Index